This policy is bullsh*t.
The girl was carrying her inhaler as well. Schools still allow kids to carry inhalers. What they don’t allow is for those kids to give their prescription inhalers to other kids to use.
If the other girl had actually been asthmatic, she too would have had an inhaler.
My son had an inhaler for a brief time while they were trying to diagnose a breathing problem. In his case, they kept the inhaler in the office, because it was not considered a life-threatening issue. It was a bit of a pain, but we got the doctor to write two prescriptions, so he could store one at the school office, and keep one at home.
According to the American Lung Association, all 50 states have laws allowing students with asthma to carry and use quick-relief asthma inhalers at school. Though the specific rules and requirements can vary by state and even school district, many prohibit sharing.